Candidate Models
While summed ordinal scales are often treated as though they are
scale data, analysed using models with a Gaussian likelihood, this
assumes that the data are continuous and unbounded, with equal distance
between measurements. This can result in model-based predictions that
are impossible under the scale in question. Instead, analysing the data
using an ordinal model allows the restrictions of the ordinal scale to
be taken into account. While this is often done with trial-level data,
we modelled the sum scores of the ordinal scale using an ordinal model
under the assumption that many of the restrictions for modelling ordinal
data still apply while using sum scores to allow for easy interpretation
of coefficients from each scale for diagnostic value. Below, we present
posterior predictive checks for several potential candidate models to
show that the ordinal model best captures the data generation process.
This is carried out for the depression data only using the same
predictors and outcomes as used in the main models reported in the
paper. The considered models are:
- The ordinal model (cumulative likelihood with a logit link): Data
arise from categorisation of a continuous latent variable.
- Beta binomial likelihood: Data are non-negative integers with a
varying probability of success from n trials arising from a Beta
distribution.
- Negative binomial likelihood: Data are non-negative integers
representing the number of failures from a sequence of i.i.d. Bernoulli
trials.
- The traditional” Gaussian likelihood: Data are unbounded and
continuous.
- Poisson likelihood: Data are non-negative integers representing the
number of events occurring in a fixed interval of time or space.
Conceptually, the ordinal and Beta binomial models are most aligned
with the data generation process. Below, dark lines represent the
observed density of the data while light lines represent model-based
predictions. A well behaving model should be able to generate the
observed data such that the shape of the two line types should be
similar.

It is apparent that the ordinal model best captures the data
generation process and thus is an appropriate model for analysing these
data. Note that predictions for the ordinal model represent the
thresholds for each ordinal category, which is roughly half of the
actual sum score.
Full Parameter Estimates
Full fixed effect parameter estimates are included for all models
below. Note that these are presented on the log scale. In the paper only
the beta terms are included along with Bayes factors evaluating evidence
in support of the point-null hypothesis relative to the alternative
hypothesis (i.e. of a non-zero effect) using the Savage-Dickey density
ratio, but not with intercepts or thresholds.
Cumulative models assume that responses come from categoristion of a
latent continuous variable. Thus, for the cumulative models there is not
a single intercept but instead the model output reports several
“intercepts” which are thresholds that partition the latent continuous
variable into response categories. Holding all other parameters
constant, these give the cumulative log-odds of responses being within a
given threshold. (To transform these into probabilities, the inverse
logit should be applied to any given threshold.) As with intercepts in
typical regression modelling, these are often not the focus of the
research question. Thus, intercepts and thresholds are omitted from the
main text but are included in full here for completeness.
Crucially, the thresholds in the cumulative models reported here are
dictated by the unique scores on the outcome. In the depression,
anxiety, and stress models each “intercept” typically represents scores
from 0 to 42 in increments of 2 as the scale requires individual scores
to be summed and multiplied by 2. However, in the short loneliness scale
used in study 1, the thresholds represent scores from 3 to 9 in
increments of 1. For the longer loneliness scale used in studies 2 and
3, the thresholds represent scores from 0 to 11 in increments of 1. In
cases where the thresholds do not match the number of unique increments
on the scale this is due to some unique values not occurring in the data
set.
As the tables are explained fully in text, no further interpretation
is given for each table.
Study 1
Main Model
|
term
|
Estimate
|
Est.Error
|
l-95% CI
|
u-95% CI
|
Rhat
|
Bulk_ESS
|
Tail_ESS
|
|
Depression
|
|
Intercept[1]
|
-4.446
|
0.364
|
-5.173
|
-3.743
|
1
|
2708
|
4429
|
|
Intercept[2]
|
-3.193
|
0.305
|
-3.811
|
-2.607
|
1
|
2786
|
4933
|
|
Intercept[3]
|
-2.503
|
0.280
|
-3.081
|
-1.976
|
1
|
2785
|
4774
|
|
Intercept[4]
|
-2.010
|
0.268
|
-2.552
|
-1.493
|
1
|
2728
|
4963
|
|
Intercept[5]
|
-1.284
|
0.253
|
-1.788
|
-0.799
|
1
|
2682
|
4818
|
|
Intercept[6]
|
-0.583
|
0.244
|
-1.061
|
-0.109
|
1
|
2829
|
5070
|
|
Intercept[7]
|
-0.117
|
0.243
|
-0.584
|
0.362
|
1
|
2968
|
5409
|
|
Intercept[8]
|
0.395
|
0.243
|
-0.071
|
0.880
|
1
|
2960
|
5350
|
|
Intercept[9]
|
0.915
|
0.246
|
0.435
|
1.402
|
1
|
3048
|
5314
|
|
Intercept[10]
|
1.372
|
0.254
|
0.882
|
1.878
|
1
|
3154
|
4487
|
|
Intercept[11]
|
1.744
|
0.259
|
1.254
|
2.262
|
1
|
3179
|
5043
|
|
Intercept[12]
|
2.193
|
0.272
|
1.679
|
2.747
|
1
|
3253
|
5292
|
|
Intercept[13]
|
2.519
|
0.280
|
1.996
|
3.093
|
1
|
3240
|
5276
|
|
Intercept[14]
|
2.756
|
0.288
|
2.206
|
3.339
|
1
|
3265
|
5019
|
|
Intercept[15]
|
3.202
|
0.305
|
2.619
|
3.829
|
1
|
3412
|
5071
|
|
Intercept[16]
|
3.926
|
0.332
|
3.293
|
4.600
|
1
|
3625
|
5399
|
|
Intercept[17]
|
4.304
|
0.350
|
3.642
|
5.009
|
1
|
3856
|
5793
|
|
Intercept[18]
|
4.824
|
0.375
|
4.095
|
5.572
|
1
|
3939
|
5591
|
|
Intercept[19]
|
5.376
|
0.414
|
4.583
|
6.207
|
1
|
4131
|
5784
|
|
Intercept[20]
|
5.972
|
0.458
|
5.096
|
6.898
|
1
|
4401
|
6349
|
|
Intercept[21]
|
6.867
|
0.546
|
5.838
|
7.998
|
1
|
5002
|
6385
|
|
Lockdown Period
|
0.451
|
0.099
|
0.257
|
0.645
|
1
|
7956
|
6747
|
|
Total Hours
|
0.360
|
0.173
|
0.021
|
0.700
|
1
|
4170
|
5539
|
|
Lockdown Period by Hours
|
-0.072
|
0.106
|
-0.275
|
0.135
|
1
|
10559
|
7063
|
|
Anxiety
|
|
Intercept[1]
|
-3.560
|
0.353
|
-4.273
|
-2.905
|
1
|
2116
|
3968
|
|
Intercept[2]
|
-1.703
|
0.284
|
-2.277
|
-1.171
|
1
|
2233
|
4186
|
|
Intercept[3]
|
-0.687
|
0.263
|
-1.204
|
-0.181
|
1
|
2483
|
4792
|
|
Intercept[4]
|
-0.051
|
0.260
|
-0.561
|
0.463
|
1
|
2453
|
4517
|
|
Intercept[5]
|
0.871
|
0.264
|
0.351
|
1.389
|
1
|
2604
|
4148
|
|
Intercept[6]
|
1.490
|
0.274
|
0.973
|
2.038
|
1
|
2676
|
4885
|
|
Intercept[7]
|
2.152
|
0.289
|
1.604
|
2.731
|
1
|
2755
|
4799
|
|
Intercept[8]
|
2.989
|
0.318
|
2.390
|
3.644
|
1
|
3003
|
4837
|
|
Intercept[9]
|
3.752
|
0.350
|
3.096
|
4.461
|
1
|
3230
|
5122
|
|
Intercept[10]
|
4.500
|
0.387
|
3.761
|
5.276
|
1
|
3600
|
5493
|
|
Intercept[11]
|
5.393
|
0.436
|
4.564
|
6.285
|
1
|
4008
|
5500
|
|
Intercept[12]
|
5.901
|
0.465
|
5.042
|
6.851
|
1
|
4271
|
5758
|
|
Intercept[13]
|
6.290
|
0.494
|
5.363
|
7.305
|
1
|
4476
|
5427
|
|
Intercept[14]
|
6.906
|
0.554
|
5.873
|
8.036
|
1
|
4912
|
5937
|
|
Intercept[15]
|
7.647
|
0.635
|
6.481
|
8.924
|
1
|
5464
|
6358
|
|
Intercept[16]
|
8.560
|
0.780
|
7.153
|
10.206
|
1
|
6382
|
5739
|
|
Intercept[17]
|
9.278
|
0.942
|
7.628
|
11.318
|
1
|
6879
|
5538
|
|
Intercept[18]
|
10.622
|
1.355
|
8.362
|
13.693
|
1
|
9188
|
6048
|
|
Lockdown Period
|
0.191
|
0.099
|
-0.002
|
0.387
|
1
|
8096
|
6130
|
|
Total Hours
|
0.333
|
0.174
|
0.001
|
0.683
|
1
|
3937
|
5259
|
|
Lockdown Period by Hours
|
-0.081
|
0.107
|
-0.295
|
0.131
|
1
|
10132
|
6493
|
|
Stress
|
|
Intercept[1]
|
-4.847
|
0.389
|
-5.655
|
-4.111
|
1
|
2445
|
3864
|
|
Intercept[2]
|
-3.920
|
0.336
|
-4.621
|
-3.284
|
1
|
2459
|
4118
|
|
Intercept[3]
|
-3.132
|
0.295
|
-3.739
|
-2.574
|
1
|
2422
|
3743
|
|
Intercept[4]
|
-2.383
|
0.265
|
-2.917
|
-1.872
|
1
|
2599
|
4366
|
|
Intercept[5]
|
-1.686
|
0.245
|
-2.180
|
-1.214
|
1
|
2670
|
4433
|
|
Intercept[6]
|
-0.982
|
0.233
|
-1.450
|
-0.528
|
1
|
2786
|
4657
|
|
Intercept[7]
|
-0.228
|
0.227
|
-0.679
|
0.222
|
1
|
2885
|
4379
|
|
Intercept[8]
|
0.433
|
0.227
|
-0.004
|
0.877
|
1
|
2989
|
4804
|
|
Intercept[9]
|
0.955
|
0.231
|
0.513
|
1.423
|
1
|
3110
|
4610
|
|
Intercept[10]
|
1.551
|
0.238
|
1.089
|
2.042
|
1
|
3119
|
4661
|
|
Intercept[11]
|
2.149
|
0.253
|
1.674
|
2.672
|
1
|
3249
|
4851
|
|
Intercept[12]
|
2.577
|
0.264
|
2.082
|
3.125
|
1
|
3281
|
5248
|
|
Intercept[13]
|
3.174
|
0.285
|
2.640
|
3.762
|
1
|
3468
|
5238
|
|
Intercept[14]
|
3.697
|
0.305
|
3.125
|
4.333
|
1
|
3529
|
5567
|
|
Intercept[15]
|
4.247
|
0.333
|
3.622
|
4.913
|
1
|
3656
|
4813
|
|
Intercept[16]
|
4.699
|
0.357
|
4.028
|
5.419
|
1
|
3719
|
5324
|
|
Intercept[17]
|
4.915
|
0.369
|
4.213
|
5.670
|
1
|
3694
|
5461
|
|
Intercept[18]
|
5.947
|
0.449
|
5.094
|
6.836
|
1
|
4257
|
5760
|
|
Intercept[19]
|
6.840
|
0.555
|
5.823
|
7.983
|
1
|
5261
|
5385
|
|
Intercept[20]
|
7.734
|
0.709
|
6.458
|
9.272
|
1
|
6499
|
6294
|
|
Intercept[21]
|
9.302
|
1.179
|
7.397
|
11.992
|
1
|
9224
|
6164
|
|
Lockdown Period
|
0.388
|
0.097
|
0.197
|
0.576
|
1
|
7318
|
6308
|
|
Total Hours
|
0.116
|
0.157
|
-0.195
|
0.423
|
1
|
3521
|
5316
|
|
Lockdown Period by Hours
|
0.053
|
0.104
|
-0.151
|
0.258
|
1
|
8079
|
5886
|
|
Loneliness
|
|
Intercept[1]
|
-2.531
|
0.288
|
-3.115
|
-1.998
|
1
|
2953
|
4888
|
|
Intercept[2]
|
-1.133
|
0.247
|
-1.631
|
-0.666
|
1
|
2962
|
4979
|
|
Intercept[3]
|
0.126
|
0.240
|
-0.345
|
0.597
|
1
|
3129
|
5454
|
|
Intercept[4]
|
1.886
|
0.270
|
1.372
|
2.428
|
1
|
3303
|
4045
|
|
Intercept[5]
|
3.218
|
0.316
|
2.628
|
3.867
|
1
|
3614
|
5057
|
|
Intercept[6]
|
4.443
|
0.384
|
3.721
|
5.230
|
1
|
3932
|
4907
|
|
Lockdown Period
|
0.620
|
0.108
|
0.410
|
0.833
|
1
|
8316
|
5558
|
|
Total Hours
|
0.013
|
0.171
|
-0.328
|
0.341
|
1
|
5076
|
6245
|
|
Lockdown Period by Hours
|
0.073
|
0.113
|
-0.146
|
0.299
|
1
|
10314
|
6402
|
Moderation Model
|
term
|
Estimate
|
Est.Error
|
l-95% CI
|
u-95% CI
|
Rhat
|
Bulk_ESS
|
Tail_ESS
|
|
Depression
|
|
Intercept[1]
|
1.169
|
0.448
|
0.277
|
2.056
|
1
|
10173
|
6821
|
|
Intercept[2]
|
1.862
|
0.437
|
1.016
|
2.743
|
1
|
9830
|
7187
|
|
Intercept[3]
|
2.396
|
0.436
|
1.549
|
3.278
|
1
|
9306
|
7066
|
|
Intercept[4]
|
2.732
|
0.441
|
1.872
|
3.619
|
1
|
9352
|
7190
|
|
Intercept[5]
|
3.037
|
0.448
|
2.166
|
3.941
|
1
|
9303
|
7335
|
|
Intercept[6]
|
3.504
|
0.462
|
2.615
|
4.425
|
1
|
9249
|
7092
|
|
Intercept[7]
|
3.799
|
0.470
|
2.891
|
4.737
|
1
|
9190
|
6928
|
|
Intercept[8]
|
4.155
|
0.480
|
3.233
|
5.115
|
1
|
9277
|
6991
|
|
Intercept[9]
|
4.440
|
0.490
|
3.495
|
5.425
|
1
|
9380
|
6936
|
|
Intercept[10]
|
4.686
|
0.499
|
3.726
|
5.685
|
1
|
9347
|
6865
|
|
Intercept[11]
|
4.913
|
0.508
|
3.938
|
5.928
|
1
|
9325
|
6815
|
|
Intercept[12]
|
5.194
|
0.518
|
4.212
|
6.230
|
1
|
9394
|
6826
|
|
Intercept[13]
|
5.428
|
0.524
|
4.442
|
6.485
|
1
|
9528
|
7160
|
|
Intercept[14]
|
5.534
|
0.526
|
4.535
|
6.604
|
1
|
9523
|
6992
|
|
Intercept[15]
|
5.911
|
0.540
|
4.895
|
6.997
|
1
|
9460
|
6870
|
|
Intercept[16]
|
6.358
|
0.555
|
5.299
|
7.472
|
1
|
9644
|
6438
|
|
Intercept[17]
|
6.647
|
0.566
|
5.557
|
7.775
|
1
|
9858
|
6458
|
|
Intercept[18]
|
6.929
|
0.576
|
5.821
|
8.099
|
1
|
9972
|
6734
|
|
Intercept[19]
|
7.387
|
0.595
|
6.241
|
8.593
|
1
|
10210
|
6061
|
|
Intercept[20]
|
7.599
|
0.605
|
6.455
|
8.810
|
1
|
10389
|
6709
|
|
Intercept[21]
|
8.132
|
0.642
|
6.902
|
9.408
|
1
|
10616
|
6687
|
|
Difference in Hours Played
|
0.013
|
0.022
|
-0.028
|
0.057
|
1
|
10203
|
5551
|
|
Loneliness During Lockdown
|
0.699
|
0.079
|
0.550
|
0.855
|
1
|
9632
|
6605
|
|
Hours by Loneliness
|
-0.001
|
0.004
|
-0.008
|
0.006
|
1
|
10067
|
5659
|
|
Anxiety
|
|
Intercept[1]
|
0.660
|
0.432
|
-0.177
|
1.520
|
1
|
9896
|
7050
|
|
Intercept[2]
|
1.498
|
0.437
|
0.657
|
2.365
|
1
|
9594
|
6663
|
|
Intercept[3]
|
2.100
|
0.445
|
1.257
|
2.980
|
1
|
9397
|
6743
|
|
Intercept[4]
|
2.369
|
0.453
|
1.513
|
3.281
|
1
|
9295
|
7151
|
|
Intercept[5]
|
2.759
|
0.465
|
1.883
|
3.696
|
1
|
9092
|
7253
|
|
Intercept[6]
|
2.944
|
0.470
|
2.056
|
3.891
|
1
|
9030
|
7200
|
|
Intercept[7]
|
3.286
|
0.481
|
2.361
|
4.257
|
1
|
9100
|
7151
|
|
Intercept[8]
|
3.761
|
0.494
|
2.809
|
4.760
|
1
|
9122
|
7015
|
|
Intercept[9]
|
4.197
|
0.507
|
3.216
|
5.214
|
1
|
8848
|
6441
|
|
Intercept[10]
|
4.556
|
0.520
|
3.548
|
5.609
|
1
|
8996
|
6610
|
|
Intercept[11]
|
5.139
|
0.550
|
4.077
|
6.259
|
1
|
8796
|
6870
|
|
Intercept[12]
|
5.567
|
0.572
|
4.478
|
6.728
|
1
|
9499
|
6926
|
|
Intercept[13]
|
5.841
|
0.593
|
4.714
|
7.038
|
1
|
10120
|
6930
|
|
Intercept[14]
|
6.473
|
0.650
|
5.246
|
7.787
|
1
|
10264
|
6811
|
|
Intercept[15]
|
7.083
|
0.736
|
5.714
|
8.577
|
1
|
10651
|
6968
|
|
Intercept[16]
|
7.722
|
0.874
|
6.133
|
9.590
|
1
|
10737
|
6622
|
|
Intercept[17]
|
8.867
|
1.225
|
6.832
|
11.660
|
1
|
12393
|
6346
|
|
Difference in Hours Played
|
0.030
|
0.022
|
-0.012
|
0.073
|
1
|
8528
|
6298
|
|
Loneliness During Lockdown
|
0.398
|
0.076
|
0.250
|
0.551
|
1
|
9196
|
6575
|
|
Hours by Loneliness
|
-0.003
|
0.004
|
-0.010
|
0.004
|
1
|
8450
|
6130
|
|
Stress
|
|
Intercept[1]
|
0.326
|
0.449
|
-0.562
|
1.196
|
1
|
10986
|
6654
|
|
Intercept[2]
|
0.841
|
0.435
|
-0.019
|
1.694
|
1
|
11227
|
6696
|
|
Intercept[3]
|
1.195
|
0.430
|
0.361
|
2.032
|
1
|
11012
|
6872
|
|
Intercept[4]
|
1.618
|
0.432
|
0.773
|
2.467
|
1
|
10573
|
6620
|
|
Intercept[5]
|
2.010
|
0.438
|
1.162
|
2.874
|
1
|
10161
|
6885
|
|
Intercept[6]
|
2.404
|
0.446
|
1.537
|
3.288
|
1
|
9828
|
6715
|
|
Intercept[7]
|
2.761
|
0.454
|
1.878
|
3.662
|
1
|
9734
|
6706
|
|
Intercept[8]
|
3.183
|
0.464
|
2.292
|
4.102
|
1
|
9830
|
6534
|
|
Intercept[9]
|
3.436
|
0.472
|
2.520
|
4.358
|
1
|
9739
|
6527
|
|
Intercept[10]
|
3.853
|
0.484
|
2.902
|
4.807
|
1
|
9712
|
6620
|
|
Intercept[11]
|
4.203
|
0.494
|
3.242
|
5.193
|
1
|
9844
|
6827
|
|
Intercept[12]
|
4.428
|
0.500
|
3.451
|
5.418
|
1
|
9879
|
6553
|
|
Intercept[13]
|
4.938
|
0.519
|
3.928
|
5.968
|
1
|
10134
|
6701
|
|
Intercept[14]
|
5.272
|
0.528
|
4.254
|
6.313
|
1
|
10246
|
6567
|
|
Intercept[15]
|
5.589
|
0.537
|
4.560
|
6.657
|
1
|
10376
|
6467
|
|
Intercept[16]
|
5.773
|
0.545
|
4.718
|
6.854
|
1
|
10411
|
6242
|
|
Intercept[17]
|
5.976
|
0.552
|
4.919
|
7.061
|
1
|
10385
|
6607
|
|
Intercept[18]
|
6.560
|
0.583
|
5.433
|
7.726
|
1
|
10993
|
6510
|
|
Intercept[19]
|
7.312
|
0.647
|
6.088
|
8.602
|
1
|
11868
|
6512
|
|
Intercept[20]
|
8.123
|
0.778
|
6.688
|
9.743
|
1
|
13028
|
6317
|
|
Intercept[21]
|
9.574
|
1.183
|
7.615
|
12.282
|
1
|
13619
|
6416
|
|
Difference in Hours Played
|
0.025
|
0.022
|
-0.017
|
0.068
|
1
|
9362
|
6687
|
|
Loneliness During Lockdown
|
0.531
|
0.076
|
0.383
|
0.682
|
1
|
9656
|
6624
|
|
Hours by Loneliness
|
-0.002
|
0.004
|
-0.010
|
0.005
|
1
|
9492
|
5892
|
Study 2
Main Model
|
term
|
Estimate
|
Est.Error
|
l-95% CI
|
u-95% CI
|
Rhat
|
Bulk_ESS
|
Tail_ESS
|
|
Depression
|
|
Intercept[1]
|
-8.129
|
1.068
|
-10.408
|
-6.226
|
1
|
2577
|
2973
|
|
Intercept[2]
|
-5.793
|
0.802
|
-7.474
|
-4.334
|
1
|
2310
|
3417
|
|
Intercept[3]
|
-4.232
|
0.675
|
-5.668
|
-3.023
|
1
|
2282
|
3782
|
|
Intercept[4]
|
-3.061
|
0.584
|
-4.265
|
-2.001
|
1
|
2301
|
3463
|
|
Intercept[5]
|
-2.132
|
0.528
|
-3.242
|
-1.163
|
1
|
2381
|
3789
|
|
Intercept[6]
|
-1.193
|
0.487
|
-2.196
|
-0.297
|
1
|
2485
|
4045
|
|
Intercept[7]
|
-0.282
|
0.463
|
-1.201
|
0.625
|
1
|
2687
|
3790
|
|
Intercept[8]
|
0.341
|
0.456
|
-0.562
|
1.227
|
1
|
2930
|
4530
|
|
Intercept[9]
|
0.869
|
0.454
|
-0.027
|
1.765
|
1
|
3018
|
4901
|
|
Intercept[10]
|
1.208
|
0.459
|
0.319
|
2.117
|
1
|
2996
|
4677
|
|
Intercept[11]
|
1.564
|
0.469
|
0.670
|
2.497
|
1
|
3031
|
4484
|
|
Intercept[12]
|
1.714
|
0.474
|
0.813
|
2.669
|
1
|
3051
|
4491
|
|
Intercept[13]
|
2.529
|
0.512
|
1.573
|
3.562
|
1
|
3242
|
4971
|
|
Intercept[14]
|
3.320
|
0.566
|
2.271
|
4.490
|
1
|
3343
|
5008
|
|
Intercept[15]
|
4.019
|
0.612
|
2.880
|
5.288
|
1
|
3243
|
4932
|
|
Intercept[16]
|
4.657
|
0.663
|
3.443
|
6.036
|
1
|
3516
|
5085
|
|
Intercept[17]
|
5.212
|
0.696
|
3.927
|
6.654
|
1
|
3699
|
5000
|
|
Intercept[18]
|
5.719
|
0.738
|
4.346
|
7.260
|
1
|
3790
|
5367
|
|
Intercept[19]
|
6.348
|
0.792
|
4.871
|
7.992
|
1
|
3672
|
5842
|
|
Intercept[20]
|
6.737
|
0.827
|
5.194
|
8.463
|
1
|
3882
|
5903
|
|
Intercept[21]
|
8.326
|
1.069
|
6.417
|
10.610
|
1
|
4507
|
6010
|
|
Lockdown Period
|
0.175
|
0.149
|
-0.115
|
0.464
|
1
|
13193
|
5659
|
|
Total Hours
|
0.803
|
0.265
|
0.294
|
1.334
|
1
|
5216
|
5464
|
|
Lockdown Period by Hours
|
0.004
|
0.167
|
-0.324
|
0.331
|
1
|
10430
|
5660
|
|
Anxiety
|
|
Intercept[1]
|
-4.919
|
0.732
|
-6.441
|
-3.580
|
1
|
1309
|
2574
|
|
Intercept[2]
|
-2.981
|
0.596
|
-4.206
|
-1.887
|
1
|
1304
|
2416
|
|
Intercept[3]
|
-1.384
|
0.514
|
-2.433
|
-0.447
|
1
|
1335
|
2467
|
|
Intercept[4]
|
-0.075
|
0.476
|
-1.044
|
0.832
|
1
|
1355
|
2712
|
|
Intercept[5]
|
0.731
|
0.472
|
-0.193
|
1.663
|
1
|
1489
|
2949
|
|
Intercept[6]
|
1.752
|
0.497
|
0.815
|
2.737
|
1
|
1829
|
3527
|
|
Intercept[7]
|
2.119
|
0.508
|
1.162
|
3.159
|
1
|
1960
|
3663
|
|
Intercept[8]
|
2.922
|
0.545
|
1.904
|
4.067
|
1
|
2374
|
4806
|
|
Intercept[9]
|
3.470
|
0.580
|
2.406
|
4.666
|
1
|
2717
|
4695
|
|
Intercept[10]
|
3.911
|
0.613
|
2.795
|
5.213
|
1
|
2957
|
5078
|
|
Intercept[11]
|
4.669
|
0.687
|
3.399
|
6.106
|
1
|
3316
|
4680
|
|
Intercept[12]
|
5.757
|
0.780
|
4.322
|
7.396
|
1
|
3593
|
4535
|
|
Intercept[13]
|
6.071
|
0.805
|
4.622
|
7.761
|
1
|
3579
|
4912
|
|
Intercept[14]
|
7.095
|
0.916
|
5.420
|
8.998
|
1
|
3952
|
5212
|
|
Intercept[15]
|
7.691
|
0.997
|
5.879
|
9.770
|
1
|
4452
|
5935
|
|
Intercept[16]
|
9.344
|
1.382
|
6.971
|
12.310
|
1
|
5682
|
5964
|
|
Lockdown Period
|
0.074
|
0.149
|
-0.219
|
0.366
|
1
|
9433
|
6185
|
|
Total Hours
|
0.600
|
0.258
|
0.102
|
1.104
|
1
|
2958
|
4535
|
|
Lockdown Period by Hours
|
0.175
|
0.165
|
-0.150
|
0.501
|
1
|
7451
|
6102
|
|
Stress
|
|
Intercept[1]
|
-7.554
|
1.016
|
-9.681
|
-5.729
|
1
|
2328
|
3654
|
|
Intercept[2]
|
-6.072
|
0.832
|
-7.770
|
-4.539
|
1
|
2303
|
3437
|
|
Intercept[3]
|
-4.940
|
0.727
|
-6.459
|
-3.609
|
1
|
2197
|
3696
|
|
Intercept[4]
|
-3.712
|
0.637
|
-5.071
|
-2.546
|
1
|
2121
|
3600
|
|
Intercept[5]
|
-2.555
|
0.566
|
-3.741
|
-1.509
|
1
|
2052
|
3292
|
|
Intercept[6]
|
-1.770
|
0.522
|
-2.844
|
-0.798
|
1
|
2080
|
3638
|
|
Intercept[7]
|
-0.599
|
0.478
|
-1.556
|
0.311
|
1
|
2063
|
4014
|
|
Intercept[8]
|
-0.251
|
0.474
|
-1.205
|
0.654
|
1
|
2112
|
3662
|
|
Intercept[9]
|
0.485
|
0.473
|
-0.442
|
1.405
|
1
|
2189
|
3650
|
|
Intercept[10]
|
1.164
|
0.484
|
0.235
|
2.129
|
1
|
2221
|
3837
|
|
Intercept[11]
|
2.091
|
0.513
|
1.118
|
3.157
|
1
|
2236
|
3979
|
|
Intercept[12]
|
2.738
|
0.534
|
1.727
|
3.856
|
1
|
2344
|
3843
|
|
Intercept[13]
|
3.071
|
0.546
|
2.053
|
4.210
|
1
|
2446
|
3651
|
|
Intercept[14]
|
3.796
|
0.578
|
2.724
|
4.989
|
1
|
2594
|
4246
|
|
Intercept[15]
|
4.805
|
0.646
|
3.626
|
6.148
|
1
|
2835
|
4838
|
|
Intercept[16]
|
5.271
|
0.680
|
4.040
|
6.697
|
1
|
3066
|
4797
|
|
Intercept[17]
|
6.233
|
0.777
|
4.843
|
7.890
|
1
|
3614
|
5721
|
|
Intercept[18]
|
6.700
|
0.847
|
5.189
|
8.492
|
1
|
3854
|
5171
|
|
Intercept[19]
|
7.310
|
0.945
|
5.622
|
9.388
|
1
|
4172
|
5810
|
|
Intercept[20]
|
9.143
|
1.449
|
6.699
|
12.421
|
1
|
4399
|
5888
|
|
Lockdown Period
|
-0.073
|
0.142
|
-0.345
|
0.212
|
1
|
10153
|
5768
|
|
Total Hours
|
0.628
|
0.261
|
0.120
|
1.151
|
1
|
4154
|
5345
|
|
Lockdown Period by Hours
|
0.254
|
0.166
|
-0.068
|
0.581
|
1
|
8075
|
6024
|
|
Loneliness
|
|
Intercept[1]
|
-9.306
|
1.506
|
-12.656
|
-6.720
|
1
|
2008
|
3113
|
|
Intercept[2]
|
-6.705
|
1.022
|
-8.932
|
-4.906
|
1
|
2207
|
3303
|
|
Intercept[3]
|
-4.850
|
0.780
|
-6.500
|
-3.489
|
1
|
2214
|
3475
|
|
Intercept[4]
|
-2.950
|
0.599
|
-4.229
|
-1.867
|
1
|
2338
|
3776
|
|
Intercept[5]
|
-2.190
|
0.542
|
-3.332
|
-1.179
|
1
|
2363
|
3763
|
|
Intercept[6]
|
-1.181
|
0.490
|
-2.173
|
-0.250
|
1
|
2342
|
3954
|
|
Intercept[7]
|
-0.492
|
0.474
|
-1.409
|
0.434
|
1
|
2266
|
3716
|
|
Intercept[8]
|
0.851
|
0.492
|
-0.056
|
1.843
|
1
|
2124
|
3514
|
|
Intercept[9]
|
1.784
|
0.530
|
0.811
|
2.881
|
1
|
2061
|
3407
|
|
Intercept[10]
|
2.995
|
0.594
|
1.920
|
4.225
|
1
|
2096
|
3408
|
|
Intercept[11]
|
5.055
|
0.730
|
3.729
|
6.596
|
1
|
2088
|
4084
|
|
Lockdown Period
|
0.139
|
0.151
|
-0.159
|
0.437
|
1
|
10423
|
5682
|
|
Total Hours
|
0.212
|
0.259
|
-0.298
|
0.718
|
1
|
3971
|
5380
|
|
Lockdown Period by Hours
|
0.008
|
0.171
|
-0.330
|
0.341
|
1
|
8264
|
6196
|
Moderation Model
|
term
|
Estimate
|
Est.Error
|
l-95% CI
|
u-95% CI
|
Rhat
|
Bulk_ESS
|
Tail_ESS
|
|
Depression
|
|
Intercept[1]
|
-1.244
|
0.659
|
-2.596
|
-0.019
|
1
|
10853
|
5950
|
|
Intercept[2]
|
-0.069
|
0.533
|
-1.107
|
0.981
|
1
|
13733
|
6713
|
|
Intercept[3]
|
0.880
|
0.514
|
-0.111
|
1.906
|
1
|
12442
|
7269
|
|
Intercept[4]
|
1.190
|
0.517
|
0.189
|
2.236
|
1
|
11996
|
6713
|
|
Intercept[5]
|
1.673
|
0.531
|
0.655
|
2.746
|
1
|
11571
|
6694
|
|
Intercept[6]
|
2.050
|
0.546
|
1.001
|
3.142
|
1
|
11302
|
6751
|
|
Intercept[7]
|
2.340
|
0.558
|
1.282
|
3.477
|
1
|
11073
|
6564
|
|
Intercept[8]
|
2.745
|
0.573
|
1.663
|
3.898
|
1
|
10780
|
6730
|
|
Intercept[9]
|
3.106
|
0.589
|
1.988
|
4.278
|
1
|
10749
|
6856
|
|
Intercept[10]
|
3.293
|
0.596
|
2.150
|
4.486
|
1
|
10698
|
6430
|
|
Intercept[11]
|
3.487
|
0.605
|
2.329
|
4.699
|
1
|
10824
|
6582
|
|
Intercept[12]
|
3.689
|
0.614
|
2.523
|
4.925
|
1
|
10863
|
6528
|
|
Intercept[13]
|
4.051
|
0.637
|
2.844
|
5.320
|
1
|
10790
|
6298
|
|
Intercept[14]
|
4.373
|
0.657
|
3.123
|
5.676
|
1
|
10813
|
6327
|
|
Intercept[15]
|
4.546
|
0.671
|
3.264
|
5.878
|
1
|
10826
|
6406
|
|
Intercept[16]
|
4.817
|
0.690
|
3.521
|
6.191
|
1
|
11065
|
6552
|
|
Intercept[17]
|
5.226
|
0.713
|
3.874
|
6.659
|
1
|
11219
|
6580
|
|
Intercept[18]
|
5.750
|
0.755
|
4.319
|
7.279
|
1
|
11672
|
6783
|
|
Intercept[19]
|
6.090
|
0.782
|
4.607
|
7.672
|
1
|
11577
|
6981
|
|
Intercept[20]
|
7.270
|
0.988
|
5.495
|
9.400
|
1
|
12680
|
6389
|
|
Difference in Hours Played
|
-0.008
|
0.027
|
-0.061
|
0.045
|
1
|
10144
|
6248
|
|
Loneliness During Lockdown
|
0.350
|
0.070
|
0.219
|
0.488
|
1
|
12083
|
6958
|
|
Hours by Loneliness
|
0.003
|
0.003
|
-0.004
|
0.009
|
1
|
10310
|
6272
|
|
Anxiety
|
|
Intercept[1]
|
-0.313
|
0.494
|
-1.308
|
0.645
|
1
|
7965
|
6161
|
|
Intercept[2]
|
0.551
|
0.479
|
-0.367
|
1.477
|
1
|
9102
|
6689
|
|
Intercept[3]
|
1.057
|
0.486
|
0.123
|
2.012
|
1
|
8418
|
6638
|
|
Intercept[4]
|
1.653
|
0.504
|
0.683
|
2.655
|
1
|
8014
|
6817
|
|
Intercept[5]
|
2.003
|
0.520
|
1.010
|
3.033
|
1
|
7867
|
6878
|
|
Intercept[6]
|
2.305
|
0.532
|
1.271
|
3.367
|
1
|
7463
|
6810
|
|
Intercept[7]
|
2.550
|
0.542
|
1.495
|
3.629
|
1
|
7495
|
6791
|
|
Intercept[8]
|
2.885
|
0.558
|
1.802
|
4.007
|
1
|
7534
|
6502
|
|
Intercept[9]
|
3.108
|
0.571
|
2.007
|
4.271
|
1
|
7363
|
6496
|
|
Intercept[10]
|
3.440
|
0.590
|
2.315
|
4.624
|
1
|
7586
|
6775
|
|
Intercept[11]
|
3.726
|
0.609
|
2.558
|
4.963
|
1
|
7788
|
6628
|
|
Intercept[12]
|
4.061
|
0.634
|
2.870
|
5.353
|
1
|
8519
|
6741
|
|
Intercept[13]
|
4.326
|
0.659
|
3.076
|
5.632
|
1
|
8676
|
7136
|
|
Intercept[14]
|
5.320
|
0.793
|
3.857
|
6.994
|
1
|
8939
|
6931
|
|
Intercept[15]
|
6.783
|
1.187
|
4.762
|
9.435
|
1
|
10142
|
6359
|
|
Difference in Hours Played
|
0.017
|
0.026
|
-0.033
|
0.070
|
1
|
6094
|
4348
|
|
Loneliness During Lockdown
|
0.220
|
0.064
|
0.095
|
0.349
|
1
|
7270
|
5731
|
|
Hours by Loneliness
|
-0.001
|
0.003
|
-0.007
|
0.005
|
1
|
6385
|
4947
|
|
Stress
|
|
Intercept[1]
|
-1.867
|
0.635
|
-3.168
|
-0.670
|
1
|
9482
|
6356
|
|
Intercept[2]
|
-1.211
|
0.570
|
-2.341
|
-0.093
|
1
|
10822
|
6706
|
|
Intercept[3]
|
-0.680
|
0.541
|
-1.735
|
0.385
|
1
|
10998
|
6310
|
|
Intercept[4]
|
-0.012
|
0.520
|
-1.017
|
1.017
|
1
|
11805
|
6467
|
|
Intercept[5]
|
0.535
|
0.514
|
-0.456
|
1.560
|
1
|
11576
|
6547
|
|
Intercept[6]
|
0.815
|
0.514
|
-0.160
|
1.838
|
1
|
11740
|
6759
|
|
Intercept[7]
|
1.227
|
0.514
|
0.253
|
2.266
|
1
|
11677
|
6877
|
|
Intercept[8]
|
1.328
|
0.516
|
0.342
|
2.355
|
1
|
11563
|
6483
|
|
Intercept[9]
|
1.688
|
0.528
|
0.700
|
2.754
|
1
|
11382
|
6595
|
|
Intercept[10]
|
2.012
|
0.544
|
0.992
|
3.094
|
1
|
11245
|
6506
|
|
Intercept[11]
|
2.127
|
0.548
|
1.093
|
3.213
|
1
|
11215
|
6400
|
|
Intercept[12]
|
2.478
|
0.565
|
1.412
|
3.601
|
1
|
10880
|
6512
|
|
Intercept[13]
|
2.731
|
0.575
|
1.652
|
3.868
|
1
|
11107
|
6479
|
|
Intercept[14]
|
3.249
|
0.599
|
2.120
|
4.449
|
1
|
11070
|
6837
|
|
Intercept[15]
|
3.835
|
0.632
|
2.656
|
5.102
|
1
|
11353
|
6783
|
|
Intercept[16]
|
4.232
|
0.671
|
2.985
|
5.597
|
1
|
11322
|
6331
|
|
Intercept[17]
|
4.764
|
0.736
|
3.398
|
6.284
|
1
|
11337
|
6748
|
|
Intercept[18]
|
5.262
|
0.813
|
3.777
|
6.937
|
1
|
11324
|
6709
|
|
Intercept[19]
|
6.735
|
1.201
|
4.694
|
9.473
|
1
|
11764
|
6318
|
|
Difference in Hours Played
|
0.050
|
0.026
|
0.000
|
0.101
|
1
|
9122
|
6275
|
|
Loneliness During Lockdown
|
0.207
|
0.067
|
0.079
|
0.337
|
1
|
11729
|
6400
|
|
Hours by Loneliness
|
-0.004
|
0.003
|
-0.011
|
0.002
|
1
|
9086
|
6622
|
Study 3
Main Model
|
term
|
Estimate
|
Est.Error
|
l-95% CI
|
u-95% CI
|
Rhat
|
Bulk_ESS
|
Tail_ESS
|
|
Depression
|
|
Intercept[1]
|
-6.862
|
1.013
|
-9.014
|
-5.016
|
1
|
2019
|
3197
|
|
Intercept[2]
|
-4.816
|
0.774
|
-6.436
|
-3.431
|
1
|
1948
|
3181
|
|
Intercept[3]
|
-3.863
|
0.677
|
-5.283
|
-2.633
|
1
|
2001
|
3409
|
|
Intercept[4]
|
-3.062
|
0.608
|
-4.315
|
-1.964
|
1
|
1973
|
3372
|
|
Intercept[5]
|
-2.214
|
0.550
|
-3.354
|
-1.230
|
1
|
1946
|
3401
|
|
Intercept[6]
|
-1.404
|
0.508
|
-2.444
|
-0.470
|
1
|
1919
|
3687
|
|
Intercept[7]
|
-1.096
|
0.500
|
-2.122
|
-0.159
|
1
|
1961
|
3589
|
|
Intercept[8]
|
-0.133
|
0.483
|
-1.095
|
0.797
|
1
|
2040
|
3665
|
|
Intercept[9]
|
0.217
|
0.483
|
-0.739
|
1.148
|
1
|
2046
|
3867
|
|
Intercept[10]
|
0.566
|
0.485
|
-0.390
|
1.529
|
1
|
2104
|
3603
|
|
Intercept[11]
|
1.194
|
0.498
|
0.226
|
2.189
|
1
|
2129
|
3369
|
|
Intercept[12]
|
1.815
|
0.520
|
0.833
|
2.885
|
1
|
2369
|
4035
|
|
Intercept[13]
|
2.540
|
0.559
|
1.484
|
3.688
|
1
|
2478
|
3658
|
|
Intercept[14]
|
3.009
|
0.593
|
1.920
|
4.244
|
1
|
2510
|
3996
|
|
Intercept[15]
|
3.534
|
0.629
|
2.376
|
4.837
|
1
|
2720
|
4601
|
|
Intercept[16]
|
4.326
|
0.695
|
3.056
|
5.770
|
1
|
2787
|
4545
|
|
Intercept[17]
|
5.002
|
0.764
|
3.617
|
6.578
|
1
|
2982
|
4866
|
|
Intercept[18]
|
5.651
|
0.851
|
4.142
|
7.438
|
1
|
3227
|
5240
|
|
Intercept[19]
|
6.224
|
0.932
|
4.590
|
8.190
|
1
|
3435
|
5639
|
|
Intercept[20]
|
7.455
|
1.151
|
5.444
|
9.864
|
1
|
4266
|
5797
|
|
Lockdown Period
|
0.091
|
0.173
|
-0.244
|
0.429
|
1
|
7024
|
5940
|
|
Total Hours
|
0.032
|
0.365
|
-0.692
|
0.751
|
1
|
2365
|
3660
|
|
Lockdown Period by Hours
|
0.198
|
0.189
|
-0.176
|
0.573
|
1
|
7716
|
6175
|
|
Anxiety
|
|
Intercept[1]
|
-5.224
|
1.048
|
-7.506
|
-3.387
|
1
|
1139
|
2856
|
|
Intercept[2]
|
-3.532
|
0.882
|
-5.441
|
-1.981
|
1
|
1145
|
2812
|
|
Intercept[3]
|
-2.209
|
0.753
|
-3.794
|
-0.849
|
1
|
1147
|
2476
|
|
Intercept[4]
|
-1.215
|
0.696
|
-2.664
|
0.073
|
1
|
1183
|
2547
|
|
Intercept[5]
|
-0.061
|
0.657
|
-1.378
|
1.184
|
1
|
1289
|
2513
|
|
Intercept[6]
|
0.763
|
0.649
|
-0.532
|
2.025
|
1
|
1364
|
2887
|
|
Intercept[7]
|
1.256
|
0.653
|
-0.040
|
2.540
|
1
|
1450
|
2944
|
|
Intercept[8]
|
2.037
|
0.668
|
0.770
|
3.370
|
1
|
1611
|
3211
|
|
Intercept[9]
|
2.730
|
0.701
|
1.367
|
4.163
|
1
|
1791
|
3259
|
|
Intercept[10]
|
3.418
|
0.755
|
2.013
|
4.970
|
1
|
2079
|
4012
|
|
Intercept[11]
|
3.723
|
0.778
|
2.275
|
5.334
|
1
|
2180
|
3962
|
|
Intercept[12]
|
4.538
|
0.852
|
2.962
|
6.316
|
1
|
2424
|
3695
|
|
Intercept[13]
|
5.272
|
0.917
|
3.605
|
7.243
|
1
|
2702
|
4289
|
|
Intercept[14]
|
6.103
|
0.996
|
4.273
|
8.183
|
1
|
2995
|
4193
|
|
Intercept[15]
|
6.590
|
1.046
|
4.703
|
8.781
|
1
|
3217
|
4602
|
|
Intercept[16]
|
9.072
|
1.604
|
6.391
|
12.609
|
1
|
4518
|
5648
|
|
Intercept[17]
|
11.400
|
2.333
|
7.620
|
16.791
|
1
|
4801
|
4964
|
|
Lockdown Period
|
0.120
|
0.183
|
-0.239
|
0.472
|
1
|
9160
|
6052
|
|
Total Hours
|
-0.222
|
0.386
|
-0.995
|
0.521
|
1
|
2344
|
3993
|
|
Lockdown Period by Hours
|
-0.079
|
0.202
|
-0.476
|
0.318
|
1
|
8277
|
6488
|
|
Stress
|
|
Intercept[1]
|
-7.315
|
1.148
|
-9.772
|
-5.280
|
1
|
2703
|
3662
|
|
Intercept[2]
|
-6.145
|
0.980
|
-8.230
|
-4.391
|
1
|
2683
|
4218
|
|
Intercept[3]
|
-5.381
|
0.899
|
-7.287
|
-3.779
|
1
|
2705
|
4033
|
|
Intercept[4]
|
-4.321
|
0.796
|
-5.994
|
-2.897
|
1
|
2666
|
4214
|
|
Intercept[5]
|
-3.909
|
0.757
|
-5.501
|
-2.544
|
1
|
2591
|
3903
|
|
Intercept[6]
|
-2.572
|
0.644
|
-3.908
|
-1.401
|
1
|
2535
|
3861
|
|
Intercept[7]
|
-1.381
|
0.576
|
-2.546
|
-0.304
|
1
|
2502
|
4190
|
|
Intercept[8]
|
-0.805
|
0.559
|
-1.951
|
0.263
|
1
|
2549
|
4366
|
|
Intercept[9]
|
-0.057
|
0.552
|
-1.151
|
1.032
|
1
|
2653
|
4503
|
|
Intercept[10]
|
0.323
|
0.550
|
-0.769
|
1.418
|
1
|
2613
|
4235
|
|
Intercept[11]
|
0.905
|
0.559
|
-0.194
|
2.040
|
1
|
2709
|
4461
|
|
Intercept[12]
|
1.873
|
0.590
|
0.762
|
3.062
|
1
|
2916
|
4853
|
|
Intercept[13]
|
2.333
|
0.608
|
1.178
|
3.543
|
1
|
2998
|
4974
|
|
Intercept[14]
|
3.048
|
0.650
|
1.868
|
4.416
|
1
|
3096
|
5234
|
|
Intercept[15]
|
3.543
|
0.692
|
2.267
|
4.993
|
1
|
3271
|
5543
|
|
Intercept[16]
|
5.214
|
0.890
|
3.589
|
7.094
|
1
|
3713
|
5037
|
|
Intercept[17]
|
6.157
|
1.015
|
4.324
|
8.254
|
1
|
3878
|
5100
|
|
Intercept[18]
|
7.147
|
1.185
|
5.064
|
9.684
|
1
|
3889
|
4905
|
|
Intercept[19]
|
8.425
|
1.432
|
5.911
|
11.493
|
1
|
3844
|
4634
|
|
Intercept[20]
|
10.027
|
1.696
|
7.013
|
13.708
|
1
|
4369
|
5460
|
|
Lockdown Period
|
-0.002
|
0.175
|
-0.348
|
0.346
|
1
|
11602
|
5922
|
|
Total Hours
|
-0.128
|
0.353
|
-0.822
|
0.574
|
1
|
3946
|
5263
|
|
Lockdown Period by Hours
|
0.237
|
0.189
|
-0.138
|
0.613
|
1
|
9330
|
6316
|
|
Loneliness
|
|
Intercept[1]
|
-8.966
|
1.753
|
-12.941
|
-6.081
|
1
|
1977
|
3121
|
|
Intercept[2]
|
-7.584
|
1.390
|
-10.733
|
-5.204
|
1
|
1995
|
3694
|
|
Intercept[3]
|
-5.730
|
1.008
|
-7.900
|
-3.966
|
1
|
1786
|
3608
|
|
Intercept[4]
|
-4.065
|
0.768
|
-5.691
|
-2.714
|
1
|
1769
|
3810
|
|
Intercept[5]
|
-2.752
|
0.639
|
-4.090
|
-1.618
|
1
|
1619
|
3201
|
|
Intercept[6]
|
-1.673
|
0.568
|
-2.865
|
-0.620
|
1
|
1624
|
3194
|
|
Intercept[7]
|
0.037
|
0.509
|
-0.950
|
1.045
|
1
|
1903
|
3597
|
|
Intercept[8]
|
0.881
|
0.528
|
-0.102
|
1.979
|
1
|
1918
|
3521
|
|
Intercept[9]
|
1.629
|
0.564
|
0.590
|
2.797
|
1
|
2060
|
3891
|
|
Intercept[10]
|
2.927
|
0.652
|
1.738
|
4.281
|
1
|
2186
|
3992
|
|
Intercept[11]
|
5.245
|
0.858
|
3.702
|
7.082
|
1
|
2598
|
4338
|
|
Lockdown Period
|
0.153
|
0.181
|
-0.200
|
0.510
|
1
|
7045
|
5374
|
|
Total Hours
|
0.253
|
0.342
|
-0.397
|
0.949
|
1
|
2218
|
3794
|
|
Lockdown Period by Hours
|
-0.090
|
0.184
|
-0.458
|
0.268
|
1
|
6485
|
5733
|
Moderation Model
|
term
|
Estimate
|
Est.Error
|
l-95% CI
|
u-95% CI
|
Rhat
|
Bulk_ESS
|
Tail_ESS
|
|
Depression
|
|
Intercept[1]
|
-1.364
|
0.973
|
-3.397
|
0.444
|
1
|
6461
|
5301
|
|
Intercept[2]
|
0.232
|
0.757
|
-1.257
|
1.729
|
1
|
9280
|
6869
|
|
Intercept[3]
|
0.637
|
0.738
|
-0.799
|
2.095
|
1
|
9055
|
6566
|
|
Intercept[4]
|
1.094
|
0.726
|
-0.314
|
2.545
|
1
|
8996
|
6674
|
|
Intercept[5]
|
1.749
|
0.736
|
0.360
|
3.242
|
1
|
8759
|
6940
|
|
Intercept[6]
|
2.001
|
0.750
|
0.585
|
3.550
|
1
|
8695
|
6876
|
|
Intercept[7]
|
2.262
|
0.765
|
0.829
|
3.824
|
1
|
8609
|
7110
|
|
Intercept[8]
|
2.811
|
0.797
|
1.296
|
4.454
|
1
|
8394
|
6527
|
|
Intercept[9]
|
3.023
|
0.806
|
1.496
|
4.660
|
1
|
8441
|
6499
|
|
Intercept[10]
|
3.234
|
0.814
|
1.678
|
4.884
|
1
|
8278
|
6415
|
|
Intercept[11]
|
3.681
|
0.836
|
2.085
|
5.363
|
1
|
8170
|
6550
|
|
Intercept[12]
|
4.004
|
0.857
|
2.365
|
5.730
|
1
|
8059
|
6643
|
|
Intercept[13]
|
4.465
|
0.886
|
2.784
|
6.273
|
1
|
7907
|
6611
|
|
Intercept[14]
|
4.679
|
0.895
|
2.976
|
6.506
|
1
|
7854
|
6385
|
|
Intercept[15]
|
5.036
|
0.917
|
3.314
|
6.882
|
1
|
7870
|
6497
|
|
Intercept[16]
|
5.305
|
0.933
|
3.539
|
7.206
|
1
|
7861
|
6700
|
|
Intercept[17]
|
5.784
|
0.977
|
3.946
|
7.775
|
1
|
8092
|
6331
|
|
Intercept[18]
|
6.180
|
1.010
|
4.275
|
8.255
|
1
|
8362
|
6513
|
|
Intercept[19]
|
6.694
|
1.069
|
4.714
|
8.885
|
1
|
8480
|
6670
|
|
Intercept[20]
|
7.441
|
1.198
|
5.267
|
9.931
|
1
|
9116
|
6774
|
|
Difference in Hours Played
|
0.014
|
0.065
|
-0.117
|
0.139
|
1
|
5558
|
4870
|
|
Loneliness During Lockdown
|
0.406
|
0.102
|
0.207
|
0.612
|
1
|
8407
|
6330
|
|
Hours by Loneliness
|
-0.003
|
0.008
|
-0.018
|
0.012
|
1
|
5594
|
4930
|
|
Anxiety
|
|
Intercept[1]
|
0.712
|
0.723
|
-0.705
|
2.146
|
1
|
10649
|
6005
|
|
Intercept[2]
|
1.266
|
0.729
|
-0.128
|
2.738
|
1
|
11443
|
6143
|
|
Intercept[3]
|
1.752
|
0.747
|
0.340
|
3.246
|
1
|
10733
|
6512
|
|
Intercept[4]
|
2.138
|
0.762
|
0.697
|
3.649
|
1
|
10313
|
6596
|
|
Intercept[5]
|
2.365
|
0.776
|
0.904
|
3.901
|
1
|
9986
|
6553
|
|
Intercept[6]
|
2.825
|
0.808
|
1.319
|
4.417
|
1
|
9425
|
6504
|
|
Intercept[7]
|
3.055
|
0.821
|
1.512
|
4.674
|
1
|
9247
|
6597
|
|
Intercept[8]
|
3.464
|
0.841
|
1.888
|
5.146
|
1
|
9013
|
6401
|
|
Intercept[9]
|
3.828
|
0.864
|
2.216
|
5.554
|
1
|
9010
|
6355
|
|
Intercept[10]
|
4.026
|
0.874
|
2.367
|
5.749
|
1
|
9116
|
6766
|
|
Intercept[11]
|
4.242
|
0.882
|
2.587
|
6.001
|
1
|
9052
|
6526
|
|
Intercept[12]
|
4.609
|
0.900
|
2.916
|
6.418
|
1
|
8953
|
6605
|
|
Intercept[13]
|
5.082
|
0.939
|
3.303
|
6.945
|
1
|
9416
|
6693
|
|
Intercept[14]
|
5.497
|
0.975
|
3.671
|
7.473
|
1
|
9626
|
6532
|
|
Intercept[15]
|
7.229
|
1.350
|
4.836
|
10.157
|
1
|
11167
|
5835
|
|
Difference in Hours Played
|
-0.024
|
0.073
|
-0.171
|
0.115
|
1
|
5228
|
5204
|
|
Loneliness During Lockdown
|
0.312
|
0.097
|
0.126
|
0.503
|
1
|
8942
|
6273
|
|
Hours by Loneliness
|
0.002
|
0.008
|
-0.014
|
0.020
|
1
|
5310
|
5103
|
|
Stress
|
|
Intercept[1]
|
-1.281
|
0.807
|
-2.906
|
0.271
|
1
|
6995
|
6070
|
|
Intercept[2]
|
-0.452
|
0.743
|
-1.945
|
0.983
|
1
|
8609
|
6642
|
|
Intercept[3]
|
-0.023
|
0.723
|
-1.470
|
1.402
|
1
|
8587
|
6431
|
|
Intercept[4]
|
0.420
|
0.715
|
-0.995
|
1.820
|
1
|
8235
|
6887
|
|
Intercept[5]
|
0.611
|
0.712
|
-0.784
|
2.019
|
1
|
8351
|
6816
|
|
Intercept[6]
|
1.027
|
0.707
|
-0.347
|
2.423
|
1
|
8324
|
7044
|
|
Intercept[7]
|
1.251
|
0.710
|
-0.132
|
2.627
|
1
|
8185
|
6848
|
|
Intercept[8]
|
1.612
|
0.718
|
0.217
|
3.035
|
1
|
8109
|
6684
|
|
Intercept[9]
|
1.894
|
0.725
|
0.498
|
3.338
|
1
|
8095
|
6717
|
|
Intercept[10]
|
2.181
|
0.742
|
0.743
|
3.666
|
1
|
8229
|
7112
|
|
Intercept[11]
|
2.469
|
0.755
|
1.002
|
3.976
|
1
|
8026
|
6840
|
|
Intercept[12]
|
2.981
|
0.777
|
1.466
|
4.562
|
1
|
7995
|
6887
|
|
Intercept[13]
|
3.136
|
0.785
|
1.614
|
4.735
|
1
|
7960
|
6821
|
|
Intercept[14]
|
3.564
|
0.802
|
2.030
|
5.214
|
1
|
7619
|
6802
|
|
Intercept[15]
|
3.858
|
0.821
|
2.288
|
5.533
|
1
|
7432
|
6546
|
|
Intercept[16]
|
4.650
|
0.879
|
2.931
|
6.434
|
1
|
7397
|
6530
|
|
Intercept[17]
|
5.032
|
0.920
|
3.254
|
6.889
|
1
|
7749
|
6433
|
|
Intercept[18]
|
5.526
|
0.976
|
3.642
|
7.477
|
1
|
8033
|
6598
|
|
Intercept[19]
|
6.269
|
1.100
|
4.205
|
8.498
|
1
|
8661
|
6772
|
|
Difference in Hours Played
|
-0.004
|
0.076
|
-0.157
|
0.138
|
1
|
5265
|
5555
|
|
Loneliness During Lockdown
|
0.284
|
0.093
|
0.104
|
0.469
|
1
|
7989
|
6554
|
|
Hours by Loneliness
|
0.000
|
0.009
|
-0.017
|
0.017
|
1
|
5281
|
5921
|
Prior Sensitivity Check
The below plots show how sensitive the parameter estimates and Bayes
factors are to different specifications of priors. The models reported
in text had the following priors:
| Intercept |
Student-t(3, 0, 2.5) |
Student-t(3, 0, 1.5) |
Student-t(3, 0, 3) |
| Beta |
Normal(0, 1) |
Normal(0, 1) |
Normal(0, 1) |
| SD |
Exponential(1) |
Exponential(1) |
|
For each model we fitted 10 models in total (including the reported
model) with different prior specifications for the Beta (slope) terms.
For the main and moderation models these ranged from Normal(0,
0.2) to Normal(0, 2) in steps of 0.2, while for the difference
models these ranges from Normal(0, 0.1) to Normal(0,
1) in steps of 0.1.
In the plots below each model is reported showing the sensitivity of
both the parameter estimates for each term in the model and for the
Bayes factor in support of the point null (i.e. a parameter estimate of
0) relative to the alternative model (i.e. a parameter estimate of \(\neq\) 0). In all cases, the standard
deviation for the prior on the beta parameter for the models reported in
text is indicated with a vertical dotted line.
Generally, these plots show that the models are not overly sensitive
to the prior specification, giving consistent results for the parameter
estimates and Bayes factors regardless of whether the beta terms have
larger or smaller standard deviations than the models reported.
Study 1
Main Models

There is general agreement between models regardless of the magnitude
of the standard deviation for the beta prior for the parameter
estimates. While the Bayes factor generally increases as the standard
deviation for the beta prior increases, this trend is consistent and
conclusions do not change across prior specifications.
Moderation Model

As before, conclusions do not change depending upon the magnitude of
the standard deviation for the beta prior.
Study 2
Main Model

As before, conclusions do not change depending upon the magnitude of
the standard deviation for the beta prior.
Moderation Model

As before, conclusions do not change depending upon the magnitude of
the standard deviation for the beta prior.
Study 3
Main Model
As before, conclusions do not change depending upon the magnitude of
the standard deviation for the beta prior.
Moderation Model
As before, conclusions do not change depending upon the magnitude of
the standard deviation for the beta prior.
Prior Predictive Check
All plots show the density of the observed data (dark lines) plotted
against 100 random replications from the priors. In all cases relatively
informative priors were used to allow for proper inferences to be draws
using Bayes factors. Generally the priors allow for a wide range of
effects that differ from the observed data, so are thus not too
restrictive. For the main models the replications are close to the
observed data. The moderation models show that while the observed data
is plausible given the priors, the priors perhaps allow too much weight
to large effects and too little weight to middling effects. That said,
the reliability of the models is largely dependent on the posterior
predictive checks, below.
Study 1
Main Model

Moderation Model

Study 2
Main Model

Moderation Model
